Finite Group Databasebeta

← #30600311

Subgroups — #30600311

Order 15616.

984 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519834C211yesmaximal, characteristic
2C2×#16840615C212yesmaximal, complemented
3#21460993C211yesmaximal, characteristic
4#21460692C212yesmaximal, complemented
5#21454107C211yesmaximal, characteristic
6C23×#92420—21no—
7C22×C22⋊#6923—21no—
8C2×#14519834—24no—
9C2×#14519834—21no—
10C2×#14519834C2212yes—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840615—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519757C411yescharacteristic, complemented
16C2×#14519757C2211yescharacteristic
17C2×#14519756C411yescharacteristic, complemented
18#16839637—21no—
19#16839636—21no—
20#16839636—21no—
21#16839619C411yescharacteristic, complemented
22#16839619C2211yescharacteristic
23#16839617—21no—
24C61×#380C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—24no—
27C22×#92420—22no—
28C22×#92420—42no—
29C22×#92420—42no—
30C2×C22⋊#6923—24no—
31C2×C22⋊#6923—42no—
32C2×C22⋊#6923—42no—
33#14519834—24no—
34#14519834—22no—
35#14519834D414yes—
36(C2×#6923)⋊C4—22no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4—42no—
39C24×D61D411yescharacteristic
40C22×Dic122C2×C411yescharacteristic
41C23×Dic61D411yescharacteristic
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—42no—
45C22×C4×D61—42no—
46C22×C4×D61C2×C411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50#14519757—22no—
51#14519757—22no—
52#14519757—42no—
53#14519756—42no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63#14519684—21no—
64#14519684—21no—
65C23×C4×C61C2×C411yescharacteristic
66C2×C4⋊C4×C61C2×C411yescharacteristic
67C2×C4⋊C4×C61C2×C411yescharacteristic
68C2×C22⋊C4×C61—21no—
69#95×C61—21no—
70C24×C61C22⋊C411yescharacteristic
71C23×D61—22no—
72C23×D61—22no—
73C23×D61—22no—
74C23×D61—22no—
75C23×D61—41no—
76C23×D61—41no—
77C23×D61C22⋊C411yescharacteristic
78C23×D61SD1612yes—
79C23×D61Q1612yes—
80C23×D61C2×D411yescharacteristic
81C22×#6923—24no—
82C22×#6923—42no—
83C22×#6923—42no—
84C22×#6923—42no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×#6923—42no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—41no—
93C22×C4×C61C22⋊C411yescharacteristic
94C22×C4×C61C22×C411yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—22no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61C22⋊C411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—48no—
116C2×C4×D61—48no—
117C2×C4×D61C2×D412yes—
118C2×C4×D61#4112yes—
119C2×Dic122—22no—
120C2×Dic122—41no—
121C2×Dic122—41no—
122C22⋊#6923—48no—
123C2×#92420—24no—
124C2×#92420—48no—
125C2×#92420—42no—
126C2×#92420—42no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—41no—
129C22⋊C4×C61—42no—
130C22⋊Dic61—42no—
131#11758198—42no—
132#11758198—42no—
133#11758198—42no—
134#11758198—42no—
135C4⋊Dic61—42no—
136C4⋊Dic61—41no—
137C4⋊Dic61—41no—
138C4⋊Dic61—41no—
139#11758196—42no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61#9912yes—
145C23×C61#10312yes—
146C23×C61#10412yes—
147C23×C61C2×C22⋊C411yescharacteristic
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—24no—
153C22×D61—24no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—24no—
161C22×D61—24no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—44no—
165C22×D61—44no—
166C22×D61—44no—
167C22×D61—44no—
168C22×D61#10312yes—
169C22×D61#12112yes—
170C22×D61C2×SD1611yescharacteristic
171C22×D61C2×Q1611yescharacteristic
172C22×D61#13512yes—
173C2×#6923—44no—
174C2×#6923—44no—
175C2×#6923—44no—
176C2×#6923—82no—
177C2×#6923—82no—
178C2×#6923—82no—
179C2×#6923—82no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—44no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61C4×D412yes—
193C2×Dic61—24no—
194C2×Dic61—24no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61#9912yes—
206C2×Dic61#12011yescharacteristic
207C2×Dic61#12311yescharacteristic
208C4×D61—42no—
209C4×D61—42no—
210C4×D61—44no—
211C4×D61—44no—
212C4×D61—42no—
213C4×D61—42no—
214Dic122—82no—
215#92420—44no—
216#9111—611nomaximal, Sylow
217C22×C61—22no—
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61#32814yescomplemented
224C22×C61#38611yescharacteristic
225C22×C61C2×#9911yescharacteristic
226C22×C61C2×#9911yescharacteristic
227C22×C61C2×#10311yescharacteristic
228C22×C61#41611yescharacteristic
229C22×C61C2×#10411yescharacteristic
230C22×C61#42111yescharacteristic
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—21no—
238C2×D61—21no—
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—48no—
243C2×D61—48no—
244C2×D61—41no—
245C2×D61—41no—
246C2×D61—41no—
247C2×D61—41no—
248C2×D61#46512yes—
249C2×D61#46712yes—
250#6923—84no—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—81no—
254C4×C61—81no—
255Dic61—21no—
256Dic61—22no—
257Dic61—21no—
258Dic61—81no—
259Dic61—81no—
260C22×C22⋊C8—611no—
261#2848—611no—
262C2×#328—612no—
263C22⋊#95—611no—
264#1294—612no—
265C2×C61—24no—
266C2×C61C2×#32811yescharacteristic
267C2×C61#150111yescharacteristic
268C2×C61#151611yescharacteristic
269C2×C61#152011yescharacteristic
270C2×C61#187111yescharacteristic
271C2×C61#189211yescharacteristic
272C2×C61#189811yescharacteristic
273D61—22no—
274D61—22no—
275D61—42no—
276C24×C4—611no—
277C23×C8—1221no—
278C2×C22⋊Q8—611no—
279C2×C22⋊Q8—611no—
280C2×C22⋊Q8—611no—
281C22×C22⋊C4—1221no—
282C2×C22⋊C8—612no—
283C2×C22⋊C8—1224no—
284C2×C22⋊C8—1221no—
285#385—1221no—
286#385—1221no—
287#385—1221no—
288#380—611no—
289#380—611no—
290#380#692311yescharacteristic, complemented
291#379—1221no—
292C2×#95—612no—
293C2×#95—1221no—
294#328—1224no—
295C61#911111yescharacteristic, complemented, Sylow
296C25—611no—
297C22×Q8—611no—
298C23×C4—611no—
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—1222no—
304C23×C4—1222no—
305C23×C4—1222no—
306C23×C4—2442no—
307C23×C4—2442no—
308C23×C4C2×#692311yescharacteristic
309C22×C8—1222no—
310C22×C8—1224no—
311C22×C8—2442no—
312C22×C8—2442no—
313C22⋊Q8—1222no—
314C22⋊Q8—1222no—
315C22⋊Q8—2442no—
316C22⋊Q8—2442no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4C2×#692311yescharacteristic
322C2×C4⋊C4C2×#692311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1224no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—2442no—
331C2×C22⋊C4—2442no—
332C22⋊C8—614no—
333C22⋊C8—1222no—
334C22⋊C8—1224no—
335#95—21no—
336#95—1221no—
337#95—1221no—
338#95—1222no—
339#95—1222no—
340#95—2442no—
341C24—611no—
342C24—611no—
343C24—612no—
344C24—612no—
345C24—1222no—
346C24—1222no—
347C24—1222no—
348C24—1222no—
349C24—2441no—
350C24—2441no—
351C24C22⋊#692311yescharacteristic
352C2×Q8—1222no—
353C2×Q8—2441no—
354C2×Q8—2441no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—612no—
363C22×C4—612no—
364C22×C4—1222no—
365C22×C4—1221no—
366C22×C4—1222no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—1221no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1224no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2448no—
384C22×C4—2448no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4C22⋊#692311yescharacteristic
392C22×C4C22×#692311yescharacteristic
393C2×C8—1224no—
394C2×C8—2448no—
395C2×C8—2442no—
396C2×C8—2442no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—2441no—
400C4⋊C4—2442no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2442no—
404C22⋊C4—42no—
405C22⋊C4—2442no—
406C22⋊C4—2442no—
407C22⋊C4—2442no—
408C22⋊C4—2442no—
409C22⋊C4—2448no—
410C22⋊C4—2442no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—612no—
416C23—611no—
417C23—611no—
418C23—612no—
419C23—612no—
420C23—1224no—
421C23—1224no—
422C23—1221no—
423C23—1221no—
424C23—1221no—
425C23—1221no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—1221no—
430C23—1221no—
431C23—1221no—
432C23—1224no—
433C23—1224no—
434C23—2442no—
435C23—2442no—
436C23—2444no—
437C23—2444no—
438C23—2444no—
439C23—2444no—
440C23#1451983612yes—
441C23#1451984012yes—
442C23#1451984112yes—
443C23C2×C22⋊#692311yescharacteristic
444Q8—4882no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—611no—
458C2×C4—611no—
459C2×C4—612no—
460C2×C4—1224no—
461C2×C4—1224no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—4882no—
467C2×C4—4882no—
468C2×C4—4882no—
469C2×C4—4882no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2444no—
473C2×C4—2444no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2444no—
477C2×C4—2444no—
478C2×C4—2444no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4D4×#692312yes—
486C8—2444no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—612no—
494C22—612no—
495C22—1221no—
496C22—1221no—
497C22—1222no—
498C22—1222no—
499C22—1221no—
500C22—1221no—
501C22—1222no—
502C22—1222no—
503C22—1222no—
504C22—1222no—
505C22—1222no—
506C22—2448no—
507C22—2448no—
508C22—2441no—
509C22—2441no—
510C22—2441no—
511C22—2441no—
512C22#1684061514yescomplemented
513C22#1684071011yescharacteristic
514C22D61⋊#9911yescharacteristic
515C22C2×#1451983611yescharacteristic
516C22C2×#1451984011yescharacteristic
517C22#1684075211yescharacteristic
518C22C2×#1451984111yescharacteristic
519C22#1684075911yescharacteristic
520C4—22no—
521C4—41no—
522C4—81no—
523C4—81no—
524C4—1221no—
525C4—1222no—
526C4—1221no—
527C4—4884no—
528C4—4881no—
529C4—4881no—
530C2—24no—
531C2—1222no—
532C2—1222no—
533C2—2442no—
534C2C2×#1684061511yescharacteristic
535C2#2146119511yescharacteristic
536C2#2146124711yescharacteristic
537C2#2146125711yescharacteristic
538C2#2146180211yescharacteristic
539C2#2146183211yescharacteristic
540C2#2146184111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenter#380#6923
hypocenterC61#9111
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16840615C2semidirect
4. #21460692C2semidirect
15. C2×#14519757C4semidirect
17. C2×#14519756C4semidirect
21. #16839619C4semidirect
24. C61×#380C4semidirect
223. C22×C61#328semidirect
290. #380#6923semidirect
295. C61#9111semidirect
512. C22#16840615semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145198342
C2×#168406152
#214609932
#214606922
#214541072
#911161